While I have not found a definitive answer to your question of who coined the phrase 'question authority’, here are some useful links to help you. Generally, the phrase has been attributed to the baby boomer generation, Timothy Leary, and a bumper sticker summation of Socrates' teachings.
This article: http://www.monthlyreview.org/0901epstein.htm provides detailed history of anarchist/antiauthoritarian trends in the Unites States (try using control+F and typing in authority).
For a lecture on Socrates, try this link (again, using control+F when you get there): http://lilt.ilstu.edu/drjclassics/texts/socrates.shtm.
And, for more information than you probably want on Timothy Leary, try this wikipedia link: http://en.wikipedia.org/wiki/Timothy_Leary.

The origin of the phrase is sometimes attributed to Timothy Leary who was an advocate of psychedelic drug research and a proponent of the beneficial effects of LSD. His approach to "Question Authority" was wrong. Leary died in 1996.

However, Benjamin Franklin was often quoted as saying "it is the first responsibility of every citizen to question authority." That's probably where Leary got his version of the phrase.

For those with a Christian world view, there are events, issues and circumstances in our world that may require us to consult the Good Book before compliance to certain rules, regulations, laws and ordinances. If the Good Book say differently than what the government is telling us to do, or what is OK, then we must search our hearts about how to respond (Acts 5:29). In other words, if God says it's wrong; 'it's wrong'!

In many instances Christians as individuals or groups have responded with that "Question Authority" attitude. School Districts, small and large towns, police departments, state governments, employers, and many others have been challenged by those Christians who believe that the 'powers that be', needed to be questioned. When the answer/s that they got did not conform to the U.S. Constitution, and/or Scripture, then it was taken to the proper court. There, the rights and wrongs in the case were hammered out.

Christians and non-Christians of this country need to "Question Authority", and that is exactly what is happening with the T.E.A. Parties, and the recent Town Hall events. These "Town Hall" meetings which were scheduled by members of Congress that have become shouting matches, and have become an Albatross around the politicians necks. So much so, that some politicians have walked out of their own Town Hall meeting.

Scripture tells us that God has Ordained all governments of the world. So when a government, government agency, Congress or the President make decisions that just don't jive with our Christian world view, we need to question their actions. Just because they have the "authority" to make decisions, laws, rules or ordinances, does not necessarily make the decision, law, rule or ordinance a proper one.

So whether you're a Christian or not, it's a good thing to "Question Authority"; but do it in a polite and meaningful way, but always QUESTION AUTHORITY!

The above comment offers an interesting take on the subject which, as with any information presented anywhere, each reader should evaluate carefully. It does, however contain one factual implication that should be corrected. Timothy Leary lived to be 76, which is twenty years longer than the average life expectancy of a baby born in the US in 1920. Even those of his generational cohort who lived to be sixty on average would have died before reaching 76. See page 36 of this document from the CDC. Placing his death date after a description of his drug use and advocacy implies a causal connection not borne out by the facts.

By all accounts, Leary died a happy man, respected in his chosen community, spiritually at peace, and surrounded by loved ones. While the significance of his legacy is worthy of debate, it is oversimplified at best to claim he was "wrong."